Is the normality of a subgroup dependent on which group is its parent?

Normality is relative to the parent group. This is evident in the way it is defined. For a group $G$ and subgroup $N$ of $G$ we say that $N$ is normal in $G$ if for all $g \in G$ we have $N^g=N$. What brings the parent group in the definition is the "if for all $g \in G$" bit.

To give a concrete example: every group is a normal subgroup of itself, clearly. Now let $G$ be a non-abelian finite simple group and let $H$ be a proper and non-trivial subgroup of $G$. Then $H$ is normal in itself, but it is not normal in $G$.